in theory

Part 1: The offset continuation equation

sergey@sep.stanford.edu

## ABSTRACTThis paper concerns amplitude-preserving kinematically equivalent offset continuation (OC) operators. I introduce a revised partial differential OC equation as a tool to build OC operators that preserve offset-dependent reflectivity in prestack processing. The method of characteristics is applied to reveal the geometric laws of the OC process. With the help of geometric (kinematic) constructions, the equation is proved to be kinematically valid for all offsets and reflector dips in constant velocity media. In the OC process, the angle-dependent reflection coefficient is preserved, and the geometric spreading factor is transformed in accordance with the laws of geometric seismics independently of the reflector curvature. |

Offset continuation (OC) by definition is an operator that transforms common-offset seismic gathers from one constant offset to another (). Bagaini et al. recently identified OC with a whole family of prestack continuation operators, such as shot continuation (, ), dip moveout as a particular case of OC to zero offset, and three-dimensional azimuth moveout (). Possible practical applications of OC operators include regularizing seismic data by partial stacking prior to prestack migration () and interpolating missing data. Since dip moveout (DMO) represents a particular case of offset continuation to zero offset, the OC concept is also one of the possible approaches to DMO. Another prospective application of prestack continuation operators, pointed out recently by Fabio Rocca (personal communication), is prestack tomography-type velocity analysis.

In the theory of OC operators, two issues need to be addressed. The first
is *kinematic equivalence*. We expect seismic sections
obtained by OC to contain correctly positioned reflection
traveltime curves. The second issue is *amplitude equivalence*.
If the traveltimes are positioned correctly, it is wave
amplitudes that deserve most of our attention. Since the final outputs
of the seismic processing sequence are the
migrated sections, the kinematic equivalence of OC concerns
preserving the true geometry of seismic images, while the amplitude
equivalence addresses preserving the desired brightness of the images.
Apparently, there can be different definitions of
*amplitude-preserving* or *true-amplitude* processing. The most
commonly used one (, , , )
refers to the reflectivity preservation. According to this
definition, amplitude-preserving seismic data processing should make
the image amplitudes proportional to the reflection coefficients that
correspond to the initial constant-offset gathers. This point of view
implies that an amplitude-preserving OC operator tends to transform
offset-dependent amplitude factors, except for the reflection coefficient,
in accordance with the geometric seismic laws.

In this paper I introduce a theoretical approach to constructing different types of OC operators with respect to both kinematic equivalence and amplitude preservation.

The first part presents the theory for a revised OC differential equation. As early as in 1982, Bolondi et al. came up with the idea of describing OC as a continuous process by means of a partial differential equation (). However, their approximate differential OC operator, built on the results of Deregowski and Rocca's classic paper , turned out to fail in case of steep reflector dips or large offsets. In his famous Ph.D. thesis Dave Hale wrote:

The differences between this algorithm [DMO by Fourier transform] and previously published finite-difference DMO algorithms are analogous to the differences between frequency-wavenumber (, ) and finite-difference () algorithms for migration. For example, just as finite-difference migration algorithms require approximations that break down at steep dips, finite-difference DMO algorithms are inaccurate for large offsets and steep dips, even for constant velocity.Continuing this analogy, one can observe that both finite-difference and frequency-domain migration algorithms share a common origin: the wave equation. The new OC equation, presented in this paper and valid for all offsets and dips, can play an analogous role for offset continuation and dip moveout algorithms. The next section begins with a rigorous proof of the revised equation's kinematic validity. Since the OC process belongs to the wave type, it is appropriate to describe it by considering wavefronts (which in this case correspond to the traveltime curves) and ray trajectories (referred to in this paper as

INTRODUCING THE OFFSET CONTINUATION EQUATION Most of the contents of this paper refer to the following linear partial differential equation:

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Equation OCequation and the previously published OC equation () differ only with respect to the single term . However, this difference is substantial. As Appendix A proves, the range of validity for the approximate OC equation

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In order to prove the theoretical validity of equation OCequation for all offsets and reflector dips, I apply a simplified version of the ray method technique (, ) and obtain two equations to describe separately wavefront (traveltime) and amplitude transformation in the OC process. According to the formal ray theory, the leading term of the high-frequency asymptotics for a reflected wave, recorded on a seismogram, takes the form

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Equation eikonal describes the transformation of traveltime curve geometry in the OC process analogously to the eikonal equation in the wavefront propagation theory. Thus, what appear to be wavefronts of the wave motion described by OCequation are traveltime curves of reflected waves recorded on seismic sections. The law of amplitude transformation for high-frequency wave components, related to those wavefronts, is given by transport. In terms of the theory of partial differential equations, equation eikonal is the characteristic equation for OCequation.

Proof of kinematic equivalence

In order to prove the validity of equation eikonal, it is
convenient to transform it
to the coordinates of the initial shot gathers: *s*=*y*-*h*, *r*=*y*+*h*, and
. The transformed equation takes the
form

(7) |

Let *S* and *R* be the source and the reflection locations, and *O* be
a reflection point for that pair.
Note that the incident ray *SO* and the reflected ray
*OR* form a triangle with the basis on the offset *SR* (*l*=|*SR*|=*r*-*s*).
Let
be the angle of *SO* from the vertical axis, and be the
analogous angle of *RO* (Figure ocoray). Elementary trigonometry
(the law of sines)
gives us the following explicit relationships between the sides and
the angles of
the triangle *SOR*:

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ocoray
Reflection rays in a constant
velocity medium (a scheme).
Figure 1 |

Thus we have proved that equation SCeikonal, equivalent to eikonal, is valid in constant velocity media independently of the reflector geometry and the offset. This means that high-frequency asymptotic components of the waves, described by the OC equation, are located on the true reflection traveltime curves.

The theory of characteristics can provide other ways to prove the kinematic validity of equation eikonal, as described in (, ).

Offset continuation geometry: time rays

To study the laws of traveltime curve transformation in
the OC process, it is convenient to apply the method of
characteristics () to the eikonal-type equation eikonal. The
characteristics of eikonal (*bi*-characteristics with respect to
OCequation) are the trajectories of the high-frequency energy
propagation in the imaginary OC process. Following the formal analogy
with seismic rays, let's call those trajectories *time rays*, where
the word *time* refers to the fact that time rays describe
the traveltime transformation.
According to the theory of first-order partial differential equations,
time rays are determined by a set of ordinary
differential equations (characteristic equations) derived from
eikonal :

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The actual parameter that determines a particular time ray is the reflection point location. This important conclusion follows from the known parametric equations

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To visualize the concept of time rays, let's consider some simple analytic examples of its application to geometric analysis of the offset continuation process.

The
simplest and most important example is the case of a plane dipping
reflector. Putting
the origin of the *y* axis at the reflector plane intersection with the
surface, we can express the reflection traveltime after NMO in the form

(23) |

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Figure 2

The second example is the case of a point diffractor (the left side of Figure ococrv). Without loss of generality, the origin of the midpoint axis can be put above the diffraction point. In this case the zero-offset reflection traveltime curve has the well-known hyperbolic form

(26) |

(27) |

Figure 3

The third curious example (the right side of Figure ococrv) is the case
of a focusing
elliptic reflector. Let *y* be the center of the ellipse
and *h* be half the distance between the focuses of the ellipse. If both
focuses are on the surface, the zero-offset traveltime curve is defined by the
so-called ``DMO smile'' ():

(28) |

(29) |

Proof of amplitude equivalence This section discusses the connection between the laws of traveltime transformation and the laws of the corresponding amplitude transformation. The change of the wave amplitudes in the OC process is described by the first-order partial differential transport equation transport. The general solution of this equation can be found by applying the method of characteristics. It takes the explicit integral form

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Black et al. recently suggested a definition of true-amplitude DMO different from that of Born DMO. The difference consists of

two important components:

- 1.
*True-amplitude DMO addresses preserving the peak amplitude of the image wavelet instead of preserving its spectral density.*In the terms of this paper, the peak amplitude corresponds to the initial amplitude*A*instead of the spectral density amplitude*A*_{n}. A simple correction factor would help us take the difference between the two amplitudes into account. Multiplication by can be easily done at the NMO stage.- 2.
*Seismic sections are multiplied by time to correct for the geometric spreading factor prior to DMO (or in our case, offset continuation) processing.*

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I have introduced a partial differential equation OCequation and proved that the process described by it provides for a kinematically and dynamically equivalent offset continuation transform. Kinematic equivalence means that in constant velocity media the reflection traveltimes are transformed to their true locations on different offsets. Dynamic equivalence means that the geometric spreading term in the amplitudes of reflected waves transforms in accordance with the geometric seismics laws, while the angle-dependent reflection coefficient stays the same in the OC process. The amplitude properties of amplitude-preserving OC may find an important application in the seismic data processing connected with AVO interpretation .

The offset continuation equation can be applied directly to design OC operators of the finite-difference type. Other types of operators are related to different forms of the solutions of the OC equation. Part 2 of this paper will describe integral-type offset continuation operators based on the initial value problem associated with equation OCequation. Other important topics in the theory of offset continuation include

- Connection between OC and amplitude-preserving frequency-domain DMO
- Connection between OC and true-amplitude prestack migration
- Generalizing the OC concept to 3-D azimuth moveout (AMO)

Sergey Goldin drew my attention to the role of curvature dependence in reflected wave amplitudes. The last section of this paper partially overlaps with the contents of our joint paper ().

[paper,SEP,GEOPHYSICS,EAEG,MISC,SEGCON]

A RANGE OF VALIDITY FOR BOLONDI'S OC EQUATION

From the OC characteristic equation eikonal we can conclude that
the first-order
traveltime derivative with respect to offset decreases with a decrease
of the offset. At zero offset the derivative equals zero, as predicted by the
principle of reciprocity (reflection traveltime has to be an *
even* function of offset). Neglecting in eikonal leads to the characteristic equation

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Figure 4

B
SECOND-ORDER REFLECTION TRAVELTIME DERIVATIVES
In this appendix I derive formulas connecting second-order partial
derivatives of the reflection traveltime with the geometric properties
of the reflector in a constant velocity medium. These formulas are used in the
main text of the paper for the amplitude behavior description.
Let be the reflection traveltime from the source *s* to the
receiver *r*. Consider a formal equality

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The formulas derived in this appendix were used to get the formula

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5/9/2001